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## Pixar in a Box

### Unit 14: Lesson 1

Geometric transformations- Start here!
- Introduction to geometric transformations
- 1. Coordinate plane
- Graph points
- 2. Translation
- Laying out a scene using translation
- 3. Scaling
- Scaling items in a scene
- 4. Commutativity
- Commutative and non-commutative transformations
- 5. Rotation
- Finish your scene!
- 6. Composite transformations
- Composite transformations
- Getting to know Fran Kalal

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# 6. Composite transformations

Time to get a little mathy. Let's look at the general form for any transformation.

## Want to join the conversation?

- What is X0 and Y0? "Pick a point" as any point on the image? Because the white arrow is pointing at (0,0). I find it little bit confusing.

And when you scale about the origin by factor 4 in first case, where is the origin when it's scaled? is it (0,0) or (5,3)?

If the origin stays at (0,0) that means the translation didn't affect the origin of the object? When an object moves, its origin doesn't move with it ?

Thank you(7 votes)- It wasn't explained explicitly so I can see why you're confused, but essentially you need to know that X and Y are a set of points that make up the image. The subscript 0 just means that this is where the points are initially.

If for example we had a square instead of a ball then you could say it consists of four points, and if the square had sides of length one, these points would be S = {(0, 0), (1, 0), (0, 1), (1, 1)}. So anytime we apply an operation to this square we must apply it to each point that makes up the square. To scale the square up by a factor of two, multiply each value by two: S = {(0, 0), (2, 0), (0, 2), (2, 2)}, etc.(5 votes)

- What is x and what is y?(1 vote)
- the first class explains it in a very simple way(3 votes)

- Why are we able to change an equation by putting it in a different order, but the outcome is completely different? I know it works for subtraction and division, but I don't understand it for transformation.(2 votes)
- If we translate then scale
*from the origin*, the distance from the origin is increased, so all the points move farther, but only scale it like normal!(3 votes)

- what happens if you dialate on an x or y axix(2 votes)
- You can't dialate using a line, as the definition of dilation states!(2 votes)

- Is there any way to make these non commutative operations, commutative?

Also, wouldn't it be better to make the lower left point of the image the rotation point?(2 votes) - if you scale from the origin of the object rather than the origin of the grid, then scaling and translation operations become commutative.(2 votes)
- a quite interesting ain't it?(2 votes)
- The translate changes the-placement of ball and scale changes the-size of ball. In formula i can see that everything is right, but if scale changes only size why it changes placement too? OMG please help.(2 votes)
- I don't get why they don't commute because from what i think i am learning translations and scalling are always together(1 vote)
- pllllplppppplpl(1 vote)

## Video transcript

- Did you get final approval from the director? Congratulations. (crunch) Earlier, we saw that translation and scaling don't commute. Let's see if we can get a better understanding of what's going on using some algebra. Suppose we translate by an amount of five and x and three and y. Pick a point, x zero y zero, in the image we're translating. That point goes to a point, x one, y one, given by, x one equals x zero plus five, y one equals y zero plus three. Now, suppose we scale about the origin by a factor of four. Where does x one y one go? Let's call the point it goes to, x two y two. Scaling says, x two equals four times x one, and y two equals four times y one. Substitute our expressions for x one and y one. X two equals four times x zero plus five which equals four times x zero plus four times five, which equals four times x zero plus 20. Y two is equal to four times y zero plus three which equals four times y zero plus 12. This factor in front of x and y is four. So the effective scale factor is still four. However, the effective translation amount is 20 and x and 12 and y. For comparison, let's do the operations in the opposite order. Scale first that is. X one equals four times x 0, and y one equals four times y zero. Then translate... So algebraically, x two equals x one plus five, which equals four times x zero plus five and y two equals y one plus three which equals four times y zero plus three. Clearly the blue equations aren't the same as the red equations. But in either case, we can write the results of combining scaling and translation in the form x two equals s times x zero plus t x, and y two equals s times y zero plus t y. Where t x stands for the effective, or final translation amount in x, and t y is the effective translation amount in y. When two or more transformations are combined we call it a composite transformation. In the next exercise, you'll be ask to verify that this general form for composite transformation consisting of scales and translations always holds, no matter how many scales and translations are combined, and no matter what the order.